Interaction design for domains that involve complex abstractions can present significant challenges. This problem is particularly acute in domains where users lack effective means to conceptualise and articulate relevant abstractions. In this thesis, we investigate the use of domain-specific conceptual metaphors to address the challenge of presenting complex abstractions, using tonal harmony as an extended case study. This thesis presents a methodology for applying domain-specific conceptual metaphors to interactions designs for music. This domain involves complex abstractions where users with any degree of domain knowledge may have difficulty in articulating concepts. The methodology comprises several parts. Firstly, the thesis explores methods for systematically guiding conversation between musicians to elicit speech that describes music using conceptual metaphors. Recommendations for the most suitable methods are made. Secondly, the thesis presents a methodology for identifying image schemas and conceptual metaphors from transcriptions of conversations between musicians. The methodology covers rules for identifying source image schemas and extrapolating conceptual metaphors. Thirdly, the thesis presents a methodology for evaluating existing music interaction designs using domain-specific conceptual metaphors. We demonstrate that this approach can be used to identify potential areas for improvement as well as tensions in the design between certain tasks or abstractions. Fourthly, the thesis presents a case study for the development of a conceptual metaphor-influenced design process. In the case study, a set of materials are developed to be used by participants in the design process to facilitate the mapping of conceptual metaphors to elements of an interaction design without requiring knowledge of Conceptual Metaphor Theory. Finally, a pilot study is presented integrating the results of the conceptual metaphor-influenced design process into a consistent and useful prototype system. Compromises and refinements to the design proposals made during the design process are discussed and the resulting system design is detailed.